If f(x)=(1)/(4x²+2x+1); x R, then find the maximum value of f(x).

MathsCBSE Class 12 · Sample paper, session 2023-24 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2023-24: question 23(a)

From the official CBSE sample question paper for 2023-24, section B, question 23(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(a) · 2 marks · Very short answerApplication of Derivatives

If $f (x) = \frac{1}{4 x ^{2} + 2 x + 1}$ ; $x \in R$ , then find the maximum value of $f (x)$ .

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2023-24.

Hints

Open one hint at a time and try again before the next.

Hint 1

The fraction is largest when its denominator is smallest.

Hint 2

Complete the square in $4 x^{2} + 2 x + 1$ to find its least value.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let $g (x) = 4 x^{2} + 2 x + 1 = 4 (x^{2} + \frac{x}{2} + \frac{1}{16}) + 1 - \frac{1}{4} = 4 (x + \frac{1}{4})^{2} + \frac{3}{4} \geq \frac{3}{4}$ , with equality at $x = - \frac{1}{4}$ .
1½ marks
$f (x) = \frac{1}{g ( x )}$ is largest when $g (x)$ is smallest. So the maximum value of $f (x)$ is $\frac{1}{3/4} = \frac{4}{3}$ , attained at $x = - \frac{1}{4}$ .
½ mark

Answer: Maximum value $= \frac{4}{3}$ (at $x = - \frac{1}{4}$ )

Where the marks are

Completes the square to show $4 x^{2} + 2 x + 1 \geq \frac{3}{4}$ (1½ marks)
Concludes the maximum of $f$ is $\frac{4}{3}$ (½ mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Maxima and Minima

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2023-24: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics sample paper 2023-24: question 23(a)

Hints

Step-by-step solution

Revise the chapter

More questions from Application of Derivatives